Combination of events

Hello forumites
Ten pairs of shoes are in a closet. Four shoes are selected at random. Author wants me to find the probability that there will be at least one pairs of shoes among the four shoes selected.
Solution:-

Four shoes can be selected out of 10 pairs (20 number) in $\binom{20}{4}$ ways. Now we want to find the probability that there will be at least one pair of shoes among the four shoes selected which is equal to the probability that remains after deducting the probability of no pairs of shoes among the four shoes selected from the total probability.. So it is $1-\frac{\binom{10}{4}}{\binom{20}{4}}=0.956656$
But answer provided is $\frac{99}{323}=\frac{\binom{55}{2}}{\binom{20}{4} }$ Now which is wrong?

Re: Combination of events

Originally Posted by romsek

The problem with your answer is that by picking pairs, you remove the shoe in each pair that you don't pick from the possible shoes to pick.

Hello,
If we want to compute the probability of exactly one pair among the four selected from the total ten pairs of shoes, how it should be computed? Secondly what is the probability of two pairs among the four selected out of ten pairs of shoes?